We now present a theoretical analysis to show the dependence of \(c_{J^{\prime }}(t)\)  on the control field \(\mathcal {E}(t)\) . We write the Hamiltonian in the interaction picture without using the rotating wave approximation [1]}, [2]}, [3]}
\( \hat{H}_{I}(t)=-\left(\begin{array}{ccc}0 & \mu _{10}\mathcal {E}(t)e^{-i\omega _{01}t} & 0\\\mu _{10}\mathcal {E}(t)e^{i\omega _{01}t} & 0 & \mu _{21}\mathcal {E}(t)e^{-i\omega _{12}t}\\0 & \mu _{21}\mathcal {E}(t)e^{i\omega _{12}t} & 0\end{array}\right).\) 
